Donatella Danielli-Garofalo

Professor of Mathematics

1 765 49-41920

MATH 722

dgarofal@purdue.edu

Research Interest(s):

partial differential equations applied mathematics harmonic analysis

Dr. Danielli's research program lies at the interface of partial differential equations, calculus of variations, and geometric measure theory. More precisely, she is interested in elliptic and parabolic free boundary problems naturally arising in physics and engineering. Free boundary problems model phenomena where a conserved quantity or relation changes discontinuously across some value of the variables under consideration. The free boundary appears, for instance, as the interface between a fluid and the air, or water and ice. Dr. Danielli is interested in the study of regularity properties of the solution and of the interface in obstacle-type problems, where the distinctive feature is that the obstacle is confined to lie in a lower dimensional manifold, or the operator under consideration is of non-local nature. Another area of interest is the study of free boundary problems relevant in the theory of flame propagation.

Selected Publications

[1]	D. Danielli, N. Garofalo, and D. M. Nhieu. Integrability of the sub-Riemannian mean curvature of surfaces in the Heisenberg group. Proc. Amer. Math. Soc. , 140(3):811--821, 2012. [ bib \| DOI \| http ]
[2]	Donatella Danielli, Nicola Garofalo, and Nguyen Cong Phuc. Hardy-Sobolev type inequalities with sharp constants in Carnot-Carathéodory spaces. Potential Anal. , 34(3):223--242, 2011. [ bib \| DOI \| http ]
[3]	Donatella Danielli, Nicola Garofalo, Duy-Minh Nhieu, and Scott D. Pauls. The Bernstein problem for embedded surfaces in the Heisenberg group H 1. Indiana Univ. Math. J. , 59(2):563--594, 2010. [ bib \| DOI \| http ]
[4]	D. Danielli, N. Garofalo, and D. M. Nhieu. Sub-Riemannian calculus and monotonicity of the perimeter for graphical strips. Math. Z. , 265(3):617--637, 2010. [ bib \| DOI \| http ]
[5]	Donatella Danielli, Nicola Garofalo, and Niko Marola. Local behavior of p -harmonic Green's functions in metric spaces. Potential Anal. , 32(4):343--362, 2010. [ bib \| DOI \| http ]
[6]	Donatella Danielli, Nicola Garofalo, and Nguyen Cong Phuc. Inequalities of Hardy-Sobolev type in Carnot-Carathéodory spaces. In Sobolev spaces in mathematics. I , volume 8 of Int. Math. Ser. (N. Y.) , pages 117--151. Springer, New York, 2009. [ bib \| DOI \| http ]
[7]	D. Danielli, N. Garofalo, D. M. Nhieu, and S. D. Pauls. Instability of graphical strips and a positive answer to the Bernstein problem in the Heisenberg group H 1. J. Differential Geom. , 81(2):251--295, 2009. [ bib \| http ]
[8]	Donatella Danielli and Nicola Garofalo. Interior Cauchy-Schauder estimates for the heat flow in Carnot-Carathéodory spaces. Methods Appl. Anal. , 15(1):121--136, 2008. [ bib \| DOI \| http ]
[9]	D. Danielli, N. Garofalo, and D. M. Nhieu. A notable family of entire intrinsic minimal graphs in the Heisenberg group which are not perimeter minimizing. Amer. J. Math. , 130(2):317--339, 2008. [ bib \| DOI \| http ]
[10]	Donatella Danielli, Nicola Garofalo, and Duy-Minh Nhieu. A partial solution of the isoperimetric problem for the Heisenberg group. Forum Math. , 20(1):99--143, 2008. [ bib \| DOI \| http ]
[11]	D. Danielli, N. Garofalo, and D. M. Nhieu. Sub-Riemannian calculus on hypersurfaces in Carnot groups. Adv. Math. , 215(1):292--378, 2007. [ bib \| DOI \| http ]
[12]	Donatella Danielli, Nicola Garofalo, and Arshak Petrosyan. The sub-elliptic obstacle problem: C 1,α regularity of the free boundary in Carnot groups of step two. Adv. Math. , 211(2):485--516, 2007. [ bib \| DOI \| http ]
[13]	Luca Capogna, Donatella Danielli, Scott D. Pauls, and Jeremy T. Tyson. An introduction to the Heisenberg group and the sub-Riemannian isoperimetric problem , volume 259 of Progress in Mathematics . Birkhäuser Verlag, Basel, 2007. [ bib ]
[14]	Donatella Danielli and Arshak Petrosyan. Full regularity of the free boundary in a Bernoulli-type problem in two dimensions. Math. Res. Lett. , 13(4):667--681, 2006. [ bib \| DOI \| http ]
[15]	Donatella Danielli, Nicola Garofalo, and Duy-Minh Nhieu. Non-doubling Ahlfors measures, perimeter measures, and the characterization of the trace spaces of Sobolev functions in Carnot-Carathéodory spaces. Mem. Amer. Math. Soc. , 182(857):x+119, 2006. [ bib \| DOI \| http ]
[16]	Donatella Danielli and Marianne Korten. On the pointwise jump condition at the free boundary in the 1-phase Stefan problem. Commun. Pure Appl. Anal. , 4(2):357--366, 2005. [ bib \| DOI \| http ]
[17]	Donatella Danielli and Arshak Petrosyan. A minimum problem with free boundary for a degenerate quasilinear operator. Calc. Var. Partial Differential Equations , 23(1):97--124, 2005. [ bib \| DOI \| http ]
[18]	D. Danielli, N. Garofalo, D. M. Nhieu, and F. Tournier. The theorem of Busemann-Feller-Alexandrov in Carnot groups. Comm. Anal. Geom. , 12(4):853--886, 2004. [ bib \| http ]
[19]	Donatella Danielli, Nicola Garofalo, and Duy-Minh Nhieu. Notions of convexity in Carnot groups. Comm. Anal. Geom. , 11(2):263--341, 2003. [ bib \| DOI \| http ]
[20]	Donatella Danielli, Nicola Garofalo, and Duy-Minh Nhieu. On the best possible character of the L Q norm in some a priori estimates for non-divergence form equations in Carnot groups. Proc. Amer. Math. Soc. , 131(11):3487--3498, 2003. [ bib \| DOI \| http ]
[21]	D. Danielli, A. Petrosyan, and H. Shahgholian. A singular perturbation problem for the p -Laplace operator. Indiana Univ. Math. J. , 52(2):457--476, 2003. [ bib \| DOI \| http ]
[22]	Donatella Danielli, Nicola Garofalo, and Sandro Salsa. Variational inequalities with lack of ellipticity. I. Optimal interior regularity and non-degeneracy of the free boundary. Indiana Univ. Math. J. , 52(2):361--398, 2003. [ bib \| DOI \| http ]
[23]	Donatella Danielli and Nicola Garofalo. Properties of entire solutions of non-uniformly elliptic equations arising in geometry and in phase transitions. Calc. Var. Partial Differential Equations , 15(4):451--491, 2002. [ bib \| DOI \| http ]
[24]	Donatella Danielli, Nicola Garofalo, and Duy-Minh Nhieu. Sub-elliptic Besov spaces and the characterization of traces on lower dimensional manifolds. In Harmonic analysis and boundary value problems (Fayetteville, AR, 2000) , volume 277 of Contemp. Math. , pages 19--37. Amer. Math. Soc., Providence, RI, 2001. [ bib \| DOI \| http ]
[25]	Donatella Danielli. A Fefferman-Phong type inequality and applications to quasilinear subelliptic equations. Potential Anal. , 11(4):387--413, 1999. [ bib \| DOI \| http ]
[26]	Donatella Danielli, Nicola Garofalo, and Duy-Minh Nhieu. Trace inequalities for Carnot-Carathéodory spaces and applications. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) , 27(2):195--252 (1999), 1998. [ bib \| http ]
[27]	Donatella Danielli, Nicola Garofalo, and Duy-Minh Nhieu. Isoperimetric and trace inequalities with respect to Carnot-Carathéodory metrics. In Geometry Seminars, 1996--1997 (Italian) (Bologna) , pages 51--62. Univ. Stud. Bologna, Bologna, 1998. [ bib ]
[28]	Donatella Danielli and Nicola Garofalo. Geometric properties of solutions to subelliptic equations in nilpotent Lie groups. In Reaction diffusion systems (Trieste, 1995) , volume 194 of Lecture Notes in Pure and Appl. Math. , pages 89--105. Dekker, New York, 1998. [ bib ]
[29]	Luca Capogna, Donatella Danielli, and Nicola Garofalo. Subelliptic mollifiers and a basic pointwise estimate of Poincaré type. Math. Z. , 226(1):147--154, 1997. [ bib \| DOI \| http ]
[30]	Luca Capogna, Donatella Danielli, and Nicola Garofalo. Capacitary estimates and the local behavior of solutions of nonlinear subelliptic equations. Amer. J. Math. , 118(6):1153--1196, 1996. [ bib \| .pdf ]
[31]	Donatella Danielli. Regularity at the boundary for solutions of nonlinear subelliptic equations. Indiana Univ. Math. J. , 44(1):269--286, 1995. [ bib \| DOI \| http ]
[32]	Luca Capogna, Donatella Danielli, and Nicola Garofalo. The geometric Sobolev embedding for vector fields and the isoperimetric inequality. Comm. Anal. Geom. , 2(2):203--215, 1994. [ bib \| DOI \| http ]
[33]	Luca Capogna, Donatella Danielli, and Nicola Garofalo. An isoperimetric inequality and the geometric Sobolev embedding for vector fields. Math. Res. Lett. , 1(2):263--268, 1994. [ bib \| DOI \| http ]
[34]	L. Capogna, D. Danielli, and N. Garofalo. Subelliptic mollifiers and a characterization of Rellich and Poincaré domains. Rend. Sem. Mat. Univ. Politec. Torino , 51(4):361--386 (1994), 1993. Partial differential equations, I (Turin, 1993). [ bib ]
[35]	Luca Capogna, Donatella Danielli, and Nicola Garofalo. An embedding theorem and the Harnack inequality for nonlinear subelliptic equations. Comm. Partial Differential Equations , 18(9-10):1765--1794, 1993. [ bib \| DOI \| http ]
[36]	Luca Capogna, Donatella Danielli, and Nicola Garofalo. Embedding theorems and the Harnack inequality for solutions of nonlinear subelliptic equations. C. R. Acad. Sci. Paris Sér. I Math. , 316(8):809--814, 1993. [ bib ]
[37]	Donatella Danielli. Formules de représentation et théorèmes d'inclusion pour des opérateurs sous-elliptiques. C. R. Acad. Sci. Paris Sér. I Math. , 314(13):987--990, 1992. [ bib ]
[38]	D. Danielli. A compact embedding theorem for a class of degenerate Sobolev spaces. Rend. Sem. Mat. Univ. Politec. Torino , 49(3):399--420 (1993), 1991. [ bib ]